In question (1) of Homework 9 we used General Purpose Optimization Routine --
optim -- in R to find a
mode that is, hopefully, the global minimum of our
posterior distribution. Now we are
going to use
JAGS to analyze
our posterior distribution.
Specifically, we are going to use
rjags
(Download).
Download the R
program and the text files it uses:
JAGS_mds_nations.r
-- Program to run the Log-Normal Bayesian Model on the Nations
data.
Here is the R Program:#
#
# Wish 1971 nations similarities data
# Title: Perceived similarity of nations
#
# Description: Wish (1971) asked 18 students to rate the global
# similarity of different pairs of nations such as 'France and China' on
# a 9-point rating scale ranging from `1=very different' to `9=very
# similar'. The table shows the mean similarity ratings.
#
# Brazil 9.00 4.83 5.28 3.44 4.72 4.50 3.83 3.50 2.39 3.06 5.39 3.17
# Congo 4.83 9.00 4.56 5.00 4.00 4.83 3.33 3.39 4.00 3.39 2.39 3.50
# Cuba 5.28 4.56 9.00 5.17 4.11 4.00 3.61 2.94 5.50 5.44 3.17 5.11
# Egypt 3.44 5.00 5.17 9.00 4.78 5.83 4.67 3.83 4.39 4.39 3.33 4.28
# France 4.72 4.00 4.11 4.78 9.00 3.44 4.00 4.22 3.67 5.06 5.94 4.72
# India 4.50 4.83 4.00 5.83 3.44 9.00 4.11 4.50 4.11 4.50 4.28 4.00
# Israel 3.83 3.33 3.61 4.67 4.00 4.11 9.00 4.83 3.00 4.17 5.94 4.44
# Japan 3.50 3.39 2.94 3.83 4.22 4.50 4.83 9.00 4.17 4.61 6.06 4.28
# China 2.39 4.00 5.50 4.39 3.67 4.11 3.00 4.17 9.00 5.72 2.56 5.06
# USSR 3.06 3.39 5.44 4.39 5.06 4.50 4.17 4.61 5.72 9.00 5.00 6.67
# USA 5.39 2.39 3.17 3.33 5.94 4.28 5.94 6.06 2.56 5.00 9.00 3.56
# Yugoslavia 3.17 3.50 5.11 4.28 4.72 4.00 4.44 4.28 5.06 6.67 3.56 9.00
#
rm(list=ls(all=TRUE))
#
library(rjags)
#
load(url("https://legacy.voteview.com/k7ftp/wf1/nations.Rda"))
#
d <- 9 - nations
N <- nrow(d)
K <- ncol(d)
inits <- function() {
z <- matrix(runif(2*K, -5, 5), nrow=K, ncol=2)
z[10,1] <- z[10,2] <- z[11,2] <- NA
z[2,2] <- abs(z[2,2])
z[7,1] <- abs(z[7,1])
z[8,2] <- -abs(z[8,2])
z[9,1] <- -abs(z[9,1])
list("z"= z, "tau"=runif(1,0,10))}
#
jags <- jags.model(
url("https://legacy.voteview.com/k7ftp/wf1/JAGS_mds_model.bug"),
data = list('N' = N, 'dstar' = d), inits = inits,
n.chains = 2, n.adapt = 10000)
#
update(jags, n.iter = 30000)
#
z <- coda.samples(jags, 'z', 10000, thin = 10)
Here is the Model Analyzed by JAGS -- JAGS_mds_model.bug:
#
# MDS Model for Nations Data
#
model{
#
# Fix USSR
#
z[10,1] <- 0.000
z[10,2] <- 0.000
#
# Fix USA 2nd Dimension
#
z[11,2] <- 0.000
#
for (i in 1:N-1){
for (j in i+1:N){
#
#
dstar[i,j] ~ dlnorm(mu[i,j],tau)
mu[i,j] <- log(sqrt((z[i,1]-z[j,1])*(z[i,1]-z[j,1])+(z[i,2]-z[j,2])*(z[i,2]-z[j,2])))
}
}
#
## priors
#
tau ~ dunif(0,10)
#
for(l in 1:2){z[1,l] ~ dnorm(0.0, 0.01)}
z[2,1] ~ dnorm(0.0,0.01)
for(l in 1:2){
for(k in 3:6) {z[k,l] ~ dnorm(0.0, 0.01)}
}
z[7,2] ~ dnorm(0.0,0.01)
z[8,1] ~ dnorm(0.0,0.01)
z[9,2] ~ dnorm(0.0,0.01)
z[11,1] ~ dnorm(0.0,0.01)
for(l in 1:2){z[12,l] ~ dnorm(0.0, 0.01)}
#
# Kludge to fix rotation -- set 1st and 2nd dimension coordinates of
# 4 Nations to fix the sign flips
z[2,2] ~ dnorm(0.0,0.01)I(0,)
z[7,1] ~ dnorm(0.0,0.01)I(0,)
z[8,2] ~ dnorm(0.0,0.01)I(,0)
z[9,1] ~ dnorm(0.0,0.01)I(,0)
}
When you run JAGS_mds_nations_2.r in R it starts initializing and running JAGS.
Examine the posterior densities of the paramters, as below:
Here is a plot of the Nations from JAGS:
- Run the R program above and
put the Node Statistics Table into WORD NEATLY
FORMATTED! Paste in the Densities for Sigma,
sumllh2, and tau. Show every multi-mode coordinate density and
identify the country.
- Neatly graph the mean coordinates from your
JAGS run and turn in the R code you used to make the plot.